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CleanWave Revenue and Expense Functions



CleanWave is a company that provides eco-friendly pool cleaning services. The revenue function $R(p)$ and the expense function $E(p)$ in terms of price $p$ charged per service are given by:

$\begin{array}{l}

R(p)=100 p-2 p^2 \\

E(p)=700-10 p

\end{array}$



Based on the profit function $P(p)=-2 p^2+110 p-700$, at what price $p$ does CleanWave achieve maximum profit?



The formula to find the $x$-coordinate of the vertex for a quadratic function $a x^2+b x+c$ is $x=-\frac{b}{2 a}$.


A\$47
B\$10
C\$27.50
D\$17.50

Verified Answer :

  • Identify the profit function \(P(p) = -2p^2 + 110p - 700\) as a downward-opening parabola.
  • Locate the maximum profit at the vertex of the parabola using the formula \(p = -\frac{b}{2a}\).
  • Substitute the coefficients \(a = -2\) and \(b = 110\) into the formula to find the price \(p\).
  • Calculate the optimal price as \(p = \frac{110}{4} = 27.5\), leading to a maximum profit at \(\boxed{\$27.50}\).

Explanation

  1. Analysis of the Profit Function

The profit function \(P(p)\) represents the difference between the revenue generated and the expenses incurred by CleanWave at a service price \(p\). The given profit function is \(P(p) = -2p^2 + 110p - 700\). This is a quadratic equation of the form \(P(p) = ap^2 + bp + c\), where the coefficients are \(a = -2\), \(b = 110\), and \(c = -700\). Since the leading coefficient \(a = -2\) is negative, the graph of this function is a parabola that opens downward, meaning it reaches its maximum value at its vertex.

  1. Determining the Vertex Formula

To find the price \(p\) that maximizes the profit, we need to locate the vertex of the downward-opening parabola. For any quadratic function \(ax^2 + bx + c\), the \(x\)-coordinate of the vertex (which corresponds to \(p\) in our case) is calculated using the formula \(p = -\frac{b}{2a}\). Applying the values \(a = -2\) and \(b = 110\) from our profit function, we can determine the optimal price.

  1. Calculating the Optimal Price

Now, we substitute the coefficients \(a = -2\) and \(b = 110\) into the vertex formula:
\[ p = -\frac{110}{2 \times (-2)} \]
\[ p = -\frac{110}{-4} \]
\[ p = \frac{110}{4} = 27.5 \]
The calculation results in \(p = 27.5\). This means that charging \(\$27.50\) per service will maximize the company's profit.

  1. Conclusion

Based on the calculation of the vertex, the price that maximizes profit is \(\$27.50\). Comparing this to the given options (\(\$47, \$10, \$27.50, \$17.50\)), we confirm that \(\$27.50\) is the correct choice. The maximum profit occurs at a price of \(\boxed{\$27.50}\).

Examples

In business and economics, finding the optimal price is a common task. By modeling revenue and expenses as functions of price, companies can use calculus or algebra to determine the price that yields the highest possible profit. This helps businesses like service providers or retailers set prices that balance customer demand with operational costs, ultimately leading to more sustainable and profitable growth. For example, if a clothing brand wants to determine the best price for a new jacket, they would analyze their production costs and predicted sales volumes to find their own peak profit price.

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